Singularity for graph-directed conjugate equations indexed by a two-vertex digraph

TL;DR

This paper investigates the singularity and regularity of solutions to graph-directed conjugate equations on the unit interval, revealing conditions under which solutions are singular or smooth, especially in affine, linear fractional, and non-linear cases.

Contribution

It provides a comprehensive analysis of singularity conditions for solutions to graph-directed conjugate equations, including affine, linear fractional, and non-linear systems.

Findings

Solutions are singular in the affine case unless systems coincide.

A dichotomy exists between singularity and smoothness in linear fractional systems.

A sufficient condition for singularity is established in the non-linear setting.

Abstract

We study graph-directed conjugate functional equations on the unit interval indexed by the complete digraph with self-loops on two vertices. We focus on the singularity and regularity of the solutions for compatible systems of weak contractions. First, we show that both solutions are singular in the affine case unless the two systems coincide; second, we obtain a dichotomy between singularity and smoothness for a class of linear fractional systems; and finally, we give a sufficient condition for singularity in a non-linear setting.